Riemann and Lebesgue walk into a room and find a table covered with hundreds of US coins. How much money is there? Do they count the same amount of money? If you had 60 secs to count the money, which method would you use?
I'd use my intuition leaded by the amount of coins my eyes can see/guess.
Concerning both mathematicians, I guess there's something to see with "integrating" the amount of money. Riemann cannot count the amount of money because the function describing the amount of money is not Riemann integrable while it is Lebesgue integrable so Lebesgue count an amount of money. hehe